Can I extrude a path in x/y
You cannot. 
Extrusion is always performed n the local Z axis of a curve, particularly for 2D curves.

In 3D curves however you have a Tilt property (Ctrl+T by default) that you can use to your advantage. By tilting the curve by 90 degrees you can make the extrusion appear horizontal.

This is not advised however if you curve is mainly 2D. You will lose the ability to properly fill it, it will be hard to ensure bidimensionality, and among other things you will lose the ability to maintain constant width on tight corners

For 2D curves you are advised to use a second bezier curve as Bevel Object, this can consist of a simple horizontal line, or a narrow squished rectangle if you wish to give it some thickness.

You can then control the "width of extrusion" always from the bevel object. You can adjust on a per-vertex basis with a scaling factor using the curve Radius property (Alt+S by default).



It is advisable to maintain the bevel object width shorter than the tightest curvature radius to avoid self intersections and potentially resulting artifacts.